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Anosov Diffeomorphism


An Anosov diffeomorphism is a C^1 diffeomorphism phi of a manifold M to itself such that the tangent bundle of M is hyperbolic with respect to phi. Very few classes of Anosov diffeomorphisms are known. The best known is Arnold's cat map.

A hyperbolic linear map R^n->R^n with integer entries in the transformation matrix and determinant +/-1 is an Anosov diffeomorphism of the n-torus. Not every manifold admits an Anosov diffeomorphism. Anosov diffeomorphisms are expansive, and there are no Anosov diffeomorphisms on the circle.

It is conjectured that if phi:M->M is an Anosov diffeomorphism on a compact Riemannian manifold and the nonwandering set Omega(phi) of phi is M, then phi is topologically conjugate to a finite-to-one factor of an Anosov automorphism of a nilmanifold. It has been proved that any Anosov diffeomorphism on the n-torus is topologically conjugate to an Anosov automorphism, and also that Anosov diffeomorphisms are C^1 structurally stable.


See also

Anosov Automorphism, Anosov Map, Axiom A Diffeomorphism, Dynamical System

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References

Anosov, D. V. "Geodesic Flow on Closed Riemannian Manifolds of Negative Curvature." Trudy Mat. Inst. Steklov 90, 1-209, 1970.Smale, S. "Differentiable Dynamical Systems." Bull. Amer. Math. Soc. 73, 747-817, 1967.

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Anosov Diffeomorphism

Cite this as:

Weisstein, Eric W. "Anosov Diffeomorphism." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/AnosovDiffeomorphism.html

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